Let there be 2 such objects, whose coterminus edges are identical. Then, the volume of the tetrahedron is th of that of the parallelopiped. The expression can also be obtained using the following theorem :

Vectors are such a powerful tool in mathematics and physics, that many results can be proved very easily and intuitively.

Statement : If the diagonals of a parallelogram are congruent, then it is a rectangle.

Proof :

Let be the parallelogram. Let be the origin. So, the position vectors will be from . Diagonals are and . We have .

Parallelogram ABCD

Therefore, . By parallelogram law of vector addition, . So,

On squaring,

So,

This implies , or is perpendicular to . Hence, is a right angle. This being a parallelogram, will have all other angles equal to and hence it is a rectangle.

Statement : The diagonals of a kite are at right angles.

Proof :

Kite ABCD

Let be the kite, with as the origin. Clearly and are equal sides. In terms of vectors,

On squaring,

Canceling common terms,

Note that and are equal. So, . So,

Hence, . Hence the diagonals are perpendicular.

Statement : If in a tetrahedron, edges in each of the two pairs of opposite edges are perpendicular then the edges in the third pair are also perpendicular.

Proof:

In a tetrahedron, each triangle shares an edge with the other. Considering any 2 triangular faces, we are left with only 1 edge. The pair of common edge and the uncommon edge is said to be a pair of opposite edges. Let be a tetrahedron. So, , and are the pairs of opposite edges. Let any 2 of them be perpendicular.

Tetrahedron OABC

and . Therefore,

Expanding the brackets and then adding the equations,

Or

This gives i.e. . Hence is perpendicular to and these 2 form the third pair.